Towards Studying Non-Equilibrium Statistical Mechanics Through Dynamical Density Functional Theory (Article)

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TOWARDS STUDYING NON-EQUILIBRIUM STATISTICAL

MECHANICS THROUGH DYNAMICAL DENSITY FUNCTIONAL THEORY

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Wipsar Sunu Brams Dwandaru

Jurusan Pendidikan Fisika, FMIPA UNY, Karang Malang, Yogyakarta, 55281

*Corresponding author. Telp: 0821 60 580 833; Email:

wipsarian@yahoo.com

ABSTRACT

In this brief article, the extension of density functional theory for non-equilibrium systems is presented. Density functional theory is a powerful framework in order to study the static properties of electronic systems via a variatonal principle whereby the density (varies in 3 dimension space) holds a key role instead of the many-body wave function. Evans (Adv. Phys., 1979) fully realized the importance of this, and extended the theory for inhomogenous fluid systems. Then in an urgent need to extend the the theory for dynamical equilibrium or non-equilibrium systems, comes dynamical density functional theory. Here, the idea behind dynamical density functional theory is given based on the Smoluchowski equation.

Key Words: density functional theory, non-equilibrium systems, variational

principle, Smoluchowski equation.

1 _{Presented in the Conference on Theoretical Physics and Nonlinear Phenomena 2012 held at Yogyakarta on 20}

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INTRODUCTION

In everyday life many physical phenomena are

non-equilibrium in nature. These phenomena

includes, transport processes in biological

systems[1-3], (out-of-equilibrium) pattern

formation[4-5], weather forecast[6-8], the behavior

of a group of insects[1,9], birth and death rate[10],

the evolution of the stock market[11], so forth.

In physics, a wide variety of non-equilibrium

behavior can be observed ranging from high

energy physics[12-14], nano-scale

phenomena[15], quantum optics (laser and

maser)[12,16], radioactive decay[17], chemical

reactions[18], soft materials, e.g. sedimentation of

colloids[19,20], and liquid crystals[21,22], up to

astrophysics and cosmology[23].

There is an interesting underlying principle of

such diverse physical phenomena which is

worthwhile to be studied. This is their stochastic

behavior driven by external forces such that

non-vanishing current occurs in the systems.

Hence, it is clear that these phenomena should be

studied through statistical physics. However, we

cannot directly apply the usual equilibrium

statistical physics to non-equilibrium systems.

This is of course a consequence of the systems

being driven by some external potential which

need not be thermally activated. In equilibrium

regime, such currents are absence. That is why

the so called non-equilibrium statistical mechanics

is needed.

EQUILIBRIUM AND NON-EQUILIBRIUM

STATISTICAL MECHANICS

Equilibrium statistical mechanics deals with static

microscopic properties of a system, which is the

basis for thermodynamics. The foundation of

equilibrium statistical mechanics itself is

well-established, especially with the concept of Gibbs

ensemble. Using the latter, we may construct the

partition function of a system, and then the

thermodynamic properties of the system may be

derived.

This is in contrast with non-equilibrium statistical

mechanics. Although, abundant phenomena are

non-equilibrium, there is no consensus on the

foundation of non-equilibrium statistical

mechanics. One reason is that there is no widely

accepted definition of an ensemble. This drives

physicists to study non-equilibrium systems using

various kinetic theories. Furthermore, the

connections between these theories are still

lacking. In other words, non-equilibrium statistical

mechanics is still under construction.

There are, however, two main approaches in

order to study stochastic processes, ie.: i)

studying the trajectory a single Brownian particle,

which is govern by a stochastic differential

equation, and ii) studying the time evolution of the

probability density, ! r,! _{, that satisfies the}

Fokker-Planck equation. Especially in this article,

we use the second approach.

DENSITY FUNCTIONAL THEORY

As mentioned above, the standard method in

exploring a physical system using equilibrium

statistical mechanics is to construct the partition

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is still dependent upon the microscopic details of

the system, i.e. all of the degrees of freedom that

make up the system. Thus, if more particles are

involved in the system (many-body problem) then

the form of the partition function may become

more complicated.

An alternative way to study static equilibrium

systems without invoking directly the partition

function is through the so-called classical density

functional theory (DFT). This is a powerful

framework to study static equilibrium systems

based on a variational principle. This theory is

utilized in classical statistical physics for

describing phenomena in inhomogeneous fluids

including adsorption [24,25], freezing [26], surface

and interface behavior [27], and colloid-polymer

mixtures [28].

In general, the main objective of DFT is to

construct an approximate functional for the

intrinsic free energy ℱ _{of a classical system. In}

order to achieve this, the theory uses the concept

that ℱ _{can be written as a functional of the}

one-body density !(!), viz. ℱ_{[!(!)], where ! is the}

position coordinate. Moreover, ℱ_{[!(!)] may be}

obtained via a Legendre transform of the grand

potential functional Ω! ! , which is also a

functional of the density. A functional derivative of

the free energy with respect to density gives the

other thermodynamic quantities of the system.

This is why classical DFT is somewhat ‘simplifies’

the standard method of statistical, i.e.: rather than

determining the many-body probability density of

the system, we directly use the one-body density

which only depends upon three space variables.

However, one problem concerning any DFT is

that ℱ_{[!], is unknown, except for a special case of}

the hard rods model. That is why, most often, the

(excess) free energy is largely obtained by

various approximation. If we are not careful in

constructing an approximation for the free energy,

we may drift away from the true physical aspect of

the system. The functional might not reflect the

true behavior of the system. This resulted in the

so-called uncontrolled approximations.

TOWARDS DYNAMICAL DFT

DFT was first formulated for quantum mechanical

systems by P. Hohenberg and W. Kohn in 1964,

which is contained in [29]. DFT was fully

operational via a method introduced by Kohn and

L. J. Sham in 1965 [30]. For his part in the

development of DFT, W. Kohn together with J.

Pople received the noble prize in chemistry in

1998. Because of the simplicity in its application,

DFT is now considered as the standard method in

studying the structure of matter in chemistry.

The use of DFT is not jut limited to quantum

systems. Extension of DFT for finite systems was

formulated by Mermin [31]. This is the basis for

classical DFT, which is fully realized by Evans

[32].

The success of classical DFT in describing

various properties of equilibrium systems leads to

a natural desire to use it in attempts to treat

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appealing advantage of using DFT for dynamical

systems, i.e. it provides insights into the dynamics

of the systems on the microscopic level via

statistical mechanics.

Recently, there has been a vast number of results

coming out of DFT that are devoted to dealing

with off-equilibrium, near-equilibrium or relaxation

to equilibrium systems. The extension of classical

DFT to dynamical systems is known as dynamical

DFT (DDFT)[34-36] or time-dependent DFT

(TDDFT)[37-38].

APPLYING DYNAMICAL DFT

To our knowledge the idea of using DFT to study

dynamical systems and non-equilibrium systems

following the success of (equilibrium) classical

DFT was initially introduced around 1979 in [32],

to analyze the kinetics of spinodal decomposition

at the level of a varying time-dependent one-body

density. In this early development of the DDFT, it

is postulated [32, 33], without rigorous derivation,

that the current density of a system is

thermodynamically driven by the gradient of the

(equilibrium) chemical potential, the latter being

obtained as the functional derivative of the

(equilibrium) free energy functional ℱ _{! . In [33], a}

time-dependent one-body density, which has a

structure of a generalized Smoluchowski

equation, is derived using the aforementioned

assumption combined with the continuity equation

and the functional ℱ _{! . Formal derivations of the}

DDFT or TDDFT then follows from numerous

articles [34, 35, 37, 39] using various assumptions

and kinetic equations as starting points, e.g.

Browian motion [35], the Langevin equation [34],

hydrodynamic equation [39], and Newton’s

equation of motion [40]. Interestingly, the equation

for the time evolution of the one-body density that

is gained from this derivations has a rather similar

form [41]. DDFT has been employed to study the

dynamics of soft matters, including the glass

transition of dense colloidal suspension [42],

salvation of simple mixtures [43], ultrasoft particle

under time-varying potentials [44], relaxation of

model fluids of platelike colloidal particles [45],

and sedimentation of hard-sphere-like colloidal

particles confined in horizontal capillaries [46].

The concept of DDFT relies, as an input, on ℱ _{! ,}

which is a functional of the one-body density. The

current density, i.e. the average number of

non-equilibrium fluctuation comes from the

thermal fluctuation of the equilibrium system.

Here, one strictly use the equilibrium grand

potential functional.

CONCLUSION

We have explained in general equilibrium and

non-equilibrium statistical mechanics. We then

explain qualitatively about DFT. Finally, we

describe the extension of DFT to dynamical

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